OFFSET
1,9
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.
LINKS
EXAMPLE
2093 is the Heinz number of (9,6,4), corresponding to the multiset partition {{1,1},{1,2},{2,2}}, which can be made disconnected by removing only the part {1,2}, so a(2093) = 1.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[PrimeOmega[n]-Max@@PrimeOmega/@Select[Divisors[n], Length[csm[primeMS/@primeMS[#]]]!=1&], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 04 2018
STATUS
approved